Information Technology Reference
In-Depth Information
Exercise 8.12.
Let f
W
.0; 1/
!
IR be a continuous function. Show that if
Z
1
f.x/
dx
D
0
0
and
f.x/
0 for x
2
.0; 1/;
then f.x/
D
0 for all x
2
.0; 1/.
˘
Exercise 8.13.
(a) Assume that q is a non-negative function defined on the unit
interval Œ0; 1 and that the integral of q is small. That is, q.x/
0 for x
2
Œ0; 1
and
Z
1
q.x/
dx
;
0
where is a small positive number. We want to study the size of the set where q
is “fairly” large. To this end, consider the set
S
Df
x
2
Œ0; 1
I
q.x/
p
g
:
We will assume that q is such that S consists of a countable disjoint union of
intervals, i.e.,
[
.a
i
;b
i
/
\
.a
j
;b
j
/
D;
for i
¤
j:
S
D
Œa
i
;b
i
;
i
D
1
The length
j
S
j
of S is defined in a straightforward manner as
j
S
jD
X
i
D
0
a
i
/:
.b
i
Show that
j
S
j
p
:
(b) Solve the problem (
8.1
)-(
8.3
) in the case of
u
.x; 0/
D
f.x/
D
8:1 sin.3x/:
(c) Solve the problem (
8.12
)-(
8.14
) in the case of
v
.x; 0/
D
g.x/
D
8 sin.3x/:
(d) Compute
Z
1
.f .x/
g.x//
2
dx:
0
